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Principles of headphone design: A tutorial review
Avis, MR and Lewis, LJ
Title Principles of headphone design: A tutorial review
Authors Avis, MR and Lewis, LJ
Type Conference or Workshop Item
URL This version is available at: http://usir.salford.ac.uk/27397/
Published Date 2006
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AUDIO AT HOME – AES 21ST
UK CONFERENCE 2006
1
1.0 INTRODUCTION
Headphone modelling concerns the solution of a
number of coupled problems centring on a small
mechanical diaphragm vibrating in an enclosed space.
Frequently, the electro-acoustic transduction
mechanism is dynamic, and the transfer functions
between electrical (v,i) and mechanical (F,u)
quantities must be considered. These transfer
functions will in part depend on the acoustic load
(p,U) acting on the diaphragm, and this acoustic load
will also determine the pressure radiated by the
diaphragm which will, via a transfer function
dependant on the type of enclosure and the physiology
of the wearer, eventually impinge on the tympanic
membrane.
Whatever solution method is attempted, it is probable
that the end-user will regard the device as one which
generates acoustics pressures in response to the
application of electrical voltages. It seems sensible
therefore to commence the analysis by working
towards an evaluation of a transfer function in terms of
e
peardrum (1)
To this end, the technique of lumped element
modelling is introduced below.
2.0 LUMPED ELEMENT MODELING
A great many authors have used lumped-element (or
analogue circuit) modelling as a way of reducing the
differential equations of motion which describe
coupled electro-mechano-acoustic systems into
familiar networks of electrical components [eg 1,2,3].
The method has the advantage that fully coupled
models may be constructed rapidly, and for those
familiar with electrical networks and filters, the
analysis of the resulting circuit analogues is
straightforward and intuitive.
The most substantial disadvantage of lumped element
models is that in order to be ‘lumped’, each element
must not in itself exhibit wave behaviour. To clarify
with a mechanical example – if a vibrating diaphragm
is to be considered as a lumped mass, then all points
on that diaphragm must oscillate with identical
velocity (in magnitude and phase). Since most
mechanical and acoustic systems behave modally (in a
distributed rather than lumped fashion) within the
audio bandwidth, some might consider the technique
to be of limited utility, perhaps restricted to low-
frequency analysis. However, headphones are small
devices, and this allows the use of the technique to
illustrate the fundamental behaviours of the system
across a useful range of frequencies.
2.1 Simple driver / ear canal model
In order to illustrate generic headphone design
constraints, the selection of a dynamic transducer will
apply to most practical applications and facilitate
simple electro-mechanical modelling of the system.
Following conventional analysis the acoustic (p,U)
impedance analogue for a dynamic driver is as shown
in Figure 1.
PRINCIPLES OF HEADPHONE DESIGN – A TUTORIAL REVIEW
M.R.AVIS and L.J.KELLY
([email protected]) ([email protected])
Acoustics Research Centre, Research Institute for the Built and Human Environment,
University of Salford, UK
Headphone design using lumped-parameter analysis and analogous circuits is described. Specifically a
model of a dynamic driver in a circumaural enclosure is developed, but the approach may be applied to
other drivers and enclosure types. Models are presented which illustrate key design constraints such as
leak behaviour, open / closed back approaches, cup attenuation, cushion compliance and headband
tension. Limitations of the lumped parameter approach are discussed, following which an introduction to
analytical-modal and numerical modelling is presented.
M.R AVIS AND L. J.KELLY
AUDIO AT HOME – AES 21ST
UK CONFERENCE 2006
2
All except ZAB and ZAF are determined by the selection
of the drive unit, and these parameters (small-signal or
Thiele-Small) may be extracted using conventional
measurement techniques. By contrast the acoustic
loads are determined by the nature of the enclosure.
The simplest design case we might consider concerns
a driver in a sealed enclosure, attached rigidly to the
side of the head (Figure 2). The acoustic load will be
partially dependent on the input impedance to the ear
canal Za,0 which is itself a function of canal length,
cross-section and termination impedance. Despite
publications suggesting that the canal is in fact
roughly conical, the simplest model assumes a rigid
cylinder terminated with known impedance [4], in
which case [5]:
)tan(/
1
)tan(/
10
,
10,
0,
klSc
Zj
klScjZZ
TA
TA
A
(2)
The lumped impedance of an enclosed volume is well-
known:
2
00 c
V
P
VCA
(3)
Therefore the acoustic loads ZAB and ZAF shown in
Figure 1 can be replaced by the network shown in
Figure 3 below:
The pressure at the input to the ear canal P0 is (of
course) a function of driver velocity, which itself takes
a second-order bandpass response around a resonant
frequency determined by diaphragm mass MAD and the
series combination of suspension and backload
compliances CAD CAB / { CAD+CAB }. However, VF
Figure 1: Acoustic impedance analogue of generic dynamic driver where:
e = applied voltage
Bl = force factor (NA-1
)
SD = diaphragm area (m2)
RE = d.c. coil resistance (Ω)
LE = coil inductance (H)
MAD = moving diaphragm mass, acoustic units (kgm-4
)
CAD = suspension compliance, acoustic units (m5N
-1)
RAD = suspension resistance, acoustic units (Nsm-5
)
ZAB = acoustic backload (Pa.sm-3
)
ZAF = acoustic frontload (Pa.sm-3
)
1/{jω(LESD2/(Bl)
2}
eBl/{SD(RE+jωLE)}
ZAF
RAD (Bl)2/{RESD
2} MAD
U ZAB CAD
VB VF
ZA,0 ZA,TS1
l
Figure 2: Simple model of headphone / ear system
Figure 3: Acoustic circuit analogue for simple
model of headphone / ear system
CAB
CAF ZA,0
U
P0
PRINCIPLES OF HEADPHONE DESIGN – A TUTORIAL REVIEW
AUDIO AT HOME – AES 21ST
UK CONFERENCE 2006
3
generates a shunt compliance which low-pass filters
this pressure, suggesting this volume should be kept as
small as practicable whilst comfortably
accommodating the pinnae of users.
P0 describes the pressure at the input to the ear canal,
whilst the subjective sensation of music listening is
clearly dependant on the pressure at the tympanum.
Relating the pressures at each end of a cylindrical duct
of known termination impedance:
)sin()cos(,1
0
0
klZS
cjkl
pp
TA
l
(4)
This implies that there is a modal relation between the
pressures at each end of the ear canal, which further
modifies the input acoustics pressure (itself a function
of the modal ear canal input impedance).
2.2 Leak behaviour
Our assumption that the headphone will be rigidly
attached to the side of the head is clearly erroneous; in
most circum-aural designs the cup will be equipped
with cushions which are held against the side of the
head by headband tension. We might imagine that in
certain circumstances the seal with the head would be
imperfect, leading to leaks, as illustrated in Figure 6.
Figure 4 shows the transfer function p0 /e for two
cases – the load is assumed to be purely compliant in
case (2), and purely the input impedance to the ear
canal (no coupler volume) in case (1). The latter case
clearly shows the influence of ear canal resonances,
whereas the former (also shown by Poldy [3]) over
predicts the low frequency sensitivity. In both cases
the chosen driver parameters give a driver resonance
at about 1.5 kHz.
In Figure 5, the effect of the parallel impedances of
ear canal and shunting coupler volume can be seen, for
three values of VF. This time the magnitude response
is that of the pressure at the ear drum as a function of
drive unit input voltage. As expected, a large coupler
volume reduces sensitivity, although it could be argued
that increased sensitivity for small couplers comes at
the cost of narrow bandwidth of reproduction.
Such leaks may be modelled most simply as a branch
pipe which further diverts driver volume velocity from
the driver. Therefore the acoustic load becomes that
shown in Figure 7. Small leaks will almost certainly
exhibit lossy as well as inductively reactive behaviour,
but as a first approximation the leak may be modelled
as a lumped mass:
2
20
S
lM AL
(5)
Figure 5: | p1 /e | for three values of VF
Figure 4: | p0 /e | for (1): finite ear canal input
impedance, VF =0 and (2): VF =5cc, infinite ear
canal input impedance
Figure 6: Simple model of headphone / ear system
including leaks
VB VF
ZA,0 ZA,TS1
l
S2l2
M.R AVIS AND L. J.KELLY
AUDIO AT HOME – AES 21ST
UK CONFERENCE 2006
4
Incorporated into the analogous circuit in Figure 7,
this impedance acts as a high-pass filter (Figure 8).
This behaviour seems intuitively correct, when one
considers the spectral effect of pushing the cup more
firmly to the side of the head. The model also agrees
tolerably with headphone measurements on an
artificial head incorporating prefabricated leak ‘tubes’,
and with measurements on human users of various
head size and pinna-jaw configuration.
A first refinement of this model would place a resistive
loss in series with MAL , but as is often the case with
damping materials the precise determination of
acoustic resistance is difficult and may be best
approached by attempting to fit measured data with
the model approximation. Additionally, Poldy [3]
presents a number of formulae for holes and slits
which give suggested values for reactive and resistive
lumped components as functions of frequency, which
this author has used with mixed results.
2.3 Closed vs. open back designs
As has been shown in section 2.3, leak impedances
can significantly change the pressure response at the
users ear drum. Unfortunately the filtering effect
imposed by leaks cannot be compensated
electronically, since the spectral modifications are
highly variable between users. What is required is a
method by which inter-user leak variability is lessened
in its effect on the spectrum at the ear drum. This is
usually accomplished by instituting a large,
predetermined ‘leak’ between VF and VB and / or free
space. A simple configuration is shown in Figure 9.
As long as the (fixed, pre-determined) front / back
leakage path has small impedance compared to the
variable leaks around the cushion, the spectrum at the
ear drum can be maintained considerably more
constant between users than might otherwise be the
case. The analogue circuit now takes the form shown
in Figure 10.
Unfortunately, at the same time the sensitivity of the
device is reduced considerably. In many
circumstances, such a loss in sensitivity is tolerable
since as long as the drive unit remains linear in
operation to a sufficiently large excursion, the drop in
sound pressure level at the ear may be compensated by
increasing the magnitude of the electrical input
accordingly.
If, as in Figure 11, the open-back design lacks low-
frequency emphasis, this may be applied
electronically. The key feature of using a large, fixed,
resistive ‘leak’ (i.e. an open-back configuration) is that
in comparison with Figure 8, leak cases (1) to (3) have
a far less significant effect on the pressure at the ear.
Figure 7: Acoustic circuit analogue for simple model
of headphone / ear system including leaks
CAB
CAF ZA,0
U
P0MAL
VB
VF
ZA,0 ZA,TS1
l
S2l2
Figure 8: | p1 /e | for three leak conditions of
increasing size
Figure 9: Acoustic circuit analogue for open-back
headphone / ear system including leaks
PRINCIPLES OF HEADPHONE DESIGN – A TUTORIAL REVIEW
AUDIO AT HOME – AES 21ST
UK CONFERENCE 2006
5
An additional potential difficulty is found in the loss of
passive attenuation through the cup when using an
open-back approach. Where listening is to take place
in a noisy environment (for instance cueing in ‘live’
sound reproduction events, outside broadcasts, or for
some telephony / radio communication applications)
then the open back solution is often made
impracticable by noise masking of the signal. In such
situations, the headphone has to act simultaneously as
hearing protector and communication device; some of
the issues surrounding the design of successful hearing
protectors are considered in section 2.4.
2.4 Headphones as hearing protectors
Propagation from free space to beneath the cup can
take place via two mechanisms; leaks between
cushions and head, and by structure-borne
transmission through the cup itself. Rather than
exciting the cup into bending-wave motion, it is much
more likely that the cup itself remains stiff and
executes whole-body motion on the ‘suspension’
provided by the cushions. The mode of this oscillation
has a significant impact on sound propagation. Where
the cup ‘rocks’ (the cushion, on opposite sides of the
ear, executing out-of-phase motion) then the space
under the cup is not compressed, and structure-borne
transmission is insignificant. However if the cup
executes single-plane motion normal to the side of the
head (sometimes called the ‘breathing mode’) then the
compliant volume of air beneath executes significant
compression and rarefaction, generating acoustic
pressures.
The cup (if it executes whole-body motion) may
simply be regarded as a mass, and the cushion as a
suspension providing compliance and damping. In
addition, some authors consider separately the
compliance and damping offered by the flesh between
the cushion and the skull. The circuit analogue then
takes the form shown in Figure 12.
Transfer function P1(jω)/P0(jω) describes the
attenuation of the hearing protector. Neglecting for a
moment the leak impedances, we can see that the
system will be resonant, with the mass of the cup
bouncing on the compliance provided by the air
volume, with some damping afforded by the cushion
and flesh. The circuit may be solved to give a detailed
description of the separate effects of cushion and flesh
compliance and damping (schroter), or alternatively
for a more rapid and intuitive description these terms
may be considered as a single lumped compliance and
resistance CA1 and RA1 . Then:
1
1
20
1
)(1
1
AAF
A
AFAcupAF RCj
C
CMCj
P
P
(6)
Figure 10: Acoustic circuit analogue for open-back
headphone / ear system including leaks
Figure 11: (a) - Closed back response, no leaks;
(b) – open back response, no leaks and leak case
(1); (c) – open back response, leak case (2); (d) –
open back response, leak case (3)
CAB
CAF ZA,0
U
P0MALMA1
RA1
Figure 12: Acoustic circuit analogue for open-back
headphone / ear system including leaks
MA,leak RA,leak
MA,cup
CA,flesh
RA,cushP0
RA,flesh
CAFCA,cush
P1
M.R AVIS AND L. J.KELLY
AUDIO AT HOME – AES 21ST
UK CONFERENCE 2006
6
This perhaps does not look very useful, until we note
that the form of the circuit implies a resonant
response, with the resonance at
AcupAF
resMC
1 (7)
In which case our transfer function becomes:
1
22
1
2
2
0
1
)()(1 AAFres
A
AFres
res
RCjjC
CP
P
(8)
It is now clear that the function represents a second-
order low-pass network, flat at low frequency and
dropping off at 12dB per octave above a frequency
(9)
Damping term RA1 largely controls the amplitude at
resonance – if cushion / flesh damping is too small,
then some drop in attenuation performance in the
vicinity of ωres can be anticipated.
The full analogue was modelled with results shown in
Figure 13 for varying cup mass, and Figure 14 for
varying cushion stiffness.
It can be seen that the hypotheses drawn from the
simplified model are proven in the case of the more
complicated analogue of Figure 12. Increasing cup
mass lowers the resonant frequency (Figure 13), and
above resonance increases the attenuation. Reducing
cushion compliance raises the resonant frequency, as
well as increasing low frequency attenuation (Figure
14). The effect is limited by flesh compliance, which
effectively shunts the headphone cushion – ideally a
method would be sought to attach the cup directly to
the skull…
It might be thought that headband tension should have
an influence on ωres and hence the attenuation
performance of the headphone/protector. Headband
tension certainly has a major influence on the leak
impedance, which provides a flanking path which may
substantially obviate any benefits accruing from the
adoption of a ‘heavy’ cup. This is shown in Figure
15, where a 60g cup is effectively shunted by a leak of
comparatively small dimensions. Thus leak
impedances are a big issue for passive attenuation as
well as headphone reproduction. However, the
influence of headband tension on attenuation in the
absence of leaks rather depends on the linearity of
cushion compliance.
2.5 Headband tension, cushion compliance
and fluid-filled cushions
If the flesh/cushion system is regarded as substantially
linear, then headband tension only affects leak
performance and has no other influence. However, no
spring is linear over an infinite range of excursions, as
shown in Figure 16:
1
2 1
A
AF res
C
C
Figure 13: Attenuation of passive hearing protector
for varying cup mass
Figure 14: Attenuation of passive hearing protector
for varying cushion compliance
PRINCIPLES OF HEADPHONE DESIGN – A TUTORIAL REVIEW
AUDIO AT HOME – AES 21ST
UK CONFERENCE 2006
7
The gradient of the force/displacement curve
determines spring compliance. A linear spring
maintains constant compliance, whereas a non-linear
spring will become more stiff as the headband tension
is increased. Since both cushion and flesh compliance
can be expected to behave in a non-linear manner,
headband tension will have interesting effects on
attenuation.
We have seen in Figure 14 that decreasing compliance
will increase attenuation performance. With regard to
headband tension, this suggests the unsurprising result
that maximal tension is desired, since not only are
leaks minimized but also cushion stiffness may be
maximized, which will increase attenuation.
Unfortunately, extreme values of headband tension
rapidly become uncomfortable for users, and are likely
to lead to rejection of a product regardless of its
attenuation / reproduction performance.
In response to this dilemma, some manufacturers have
adopted fluid-filled cushions (where a gel is used
instead of a sponge element), since these have two
significant advantages. Firstly, the cushion moulds
more effectively to the side of the head than a sponge
type, meaning that leaks can be minimized without
excessive headband tensions. Secondly, as long as
sufficient tension is applied to overcome the
compliance of whatever bag is used to contain the gel,
and to hold the cup securely during normal head
movements, then the lumped stiffness of the cushion is
almost infinite. This means that flesh compliance
rather than cushion compliance is the limiting factor
in determining performance (Figure 17).
2.6 Limitations of the lumped-element
technique
The lumped parameter technique has been shown to be
effective in modelling the electro-acoustic performance
of small drivers in headphones, and also of great
utility in predicting the behaviour of headphone cups,
cushions and leaks. However, circumstances
sometimes arise where such models become less
effective – most notably where the ‘lumped’
assumption breaks down within a given element,
which starts to behave in a distributed manner.
Perhaps the most obvious example of this as applied to
headphones is in the case of the ‘compliant’ volume of
air under the cup. As is the case in room acoustics,
this volume will behave as a lumped element as long
as wavelength is long compared to enclosure
dimensions – the so-called ‘zeroth mode’. Once
wavelength decreases to approaching twice the largest
dimension of the enclosure (assuming half-wave
behaviour resulting from comparatively large
boundary impedances) then the first resonant modes of
Figure 15: Degradation of protector performance
due to small leak impedances
Figure 17: Attenuation performance of foam, and
fluid (gel) cushions
Figure 16: Force / displacement characteristic for
linear and non-linear springs
Force
Displacement
M.R AVIS AND L. J.KELLY
AUDIO AT HOME – AES 21ST
UK CONFERENCE 2006
8
the space will occur. Each mode is associated with an
eigenvalue – a resonant frequency – and a spatial
eigenfunction – which dictates the variance of pressure
and particle velocity throughout the enclosure. The
fact of this variance tells us that our lumped
assumption is no longer correct, and that as a result
the performance of the system will differ from that
expected.
Since circumaural headphones are small compared to
listening rooms, we might expect a lumped
assumption to perform well to a comparatively high
frequency. However, a point is often reached where a
more refined model is required. This may be because
the modelled performance of a headphone differs
significantly from ATF (Acoustic Test Fixture) or
MIRE (Microphone In Real Ear) measurements, or it
may be because (for instance with ANR (Active Noise
Reducing) headsets) a very accurate map of the sound
pressure under the cup is required in order to optimise
the placement of error-sensing microphones.
In the case of the input impedance to the ear canal, an
analytical distributed model was adopted from the
outset (eq 1) since a lumped assumption in this case
misses several important features. This distributed
model was easily incorporated with the lumped models
used for the remainder of the system. It is equally
possible to attempt analytical models of the resonant
behaviour of the volume of air under the headphone
cup, and to incorporate these with the overall lumped
scheme. Such models are introduced in section 3.
3.0 MODELING DISTRIBUTED PRESSURE
UNDER HEAD SHELLS 3.1 Comparison between lumped and wave
model
A simple example for the comparison of lumped-
parameter and model modelling techniques is shown
in Figure 18. Here, the cavity is reduced to a
cylindrical pressure chamber, driven by a given
velocity at one end.
The solution of the Helmholtz equation for this
boundary value problem is well known and can be
found, for example, in [6]. The solution is represented
in equation (10).
nqm
m q
qmAxrp ,,),,( (10)
We see a conventional summation of mode shapes
(eigenfunctions) of the cylinder; modes in the
longitudinal plane are represented by Ψn and modes in
the circular plane are represented by Ψm,q. Ψn form a
set of modes based on trigonometric functions which
are harmonically related, and Ψm,q form a second set
which are based on the roots of a Bessel function, and
which are not harmonically related. Am,q represents
the set of coefficients which determine the contribution
of each mode to the overall response – effectively a set
of modal ‘gain’ terms.
In the simple case of uniform excitation over the
vibrating surface with rigid cavity walls, none of
modes Ψm,q are excited and thus the response is
governed by a harmonic series of longitudinal
trigonometric functions. For unit velocity excitation,
the expression for the input impedance to the cavity is
found as:
)cot(0 kLcjZn
n (11)
foam
Ear canal wall
vibrating driver
cylinder
Figure 18: Coupling cavity simplified to cylindrical
cavity
Figure 19: lumped vs. modal model for a cylindrical
cavity
0.001 0.01 0.1 1 10 100
110
120
130
140
150
160
170
180
190
ka
|z| (d
B)
wave model
lumped model
PRINCIPLES OF HEADPHONE DESIGN – A TUTORIAL REVIEW
AUDIO AT HOME – AES 21ST
UK CONFERENCE 2006
9
This is the same result as that adopted for the input
impedance to the ear canal, expressed in equation 2. It
can be compared to a lumped model of the air in the
cavity as a simple compliance, the impedance of which
is given in equation 3. Figure 19 compares these two
formulations.
As we might expect, at high frequency the two models
show significant divergence, and this can cause
problems in the application cases previously
mentioned.
3.2 Numerical methods
A modal decomposition approach may well yield
satisfactory results for simple boundary geometries,
but a headphone cavity is rarely in practice a perfect
cylinder. In addition, its walls will exhibit complex
and frequency-dependant boundary impedances, and
the drive unit will excite the cavity over an area rather
than as a point source. The problems may be soluble,
but the complexity of the problem increases
dramatically and the modal decomposition technique
becomes less attractive.
Techniques exist that allow these problems to be
solved more generally using numerical methods, the
most common of these being the Finite Element
Method (FEM) and the Boundary Element Method
(BEM). Both FEM and BEM are based on a similar
premise that the domain to be modelled is
approximated by dividing it into small discrete
sections or elements.
The principle behind FE methods are based on
dividing the problem domain into N small elements.
These elements coexist by means of continuity
conditions, so that the behaviour of each element is
determined by the behaviour of those elements
immediately surrounding it. Elements which border a
boundary or source are given relevant boundary
conditions. In this manner we divide the problem into
that of N small problems. The numerical scheme
behind FEM requires the partial differential equation
(PDE) governing each element to be rewritten into its
‘weak form’, and an approximation to the solution is
developed and expressed as a sum of piecewise linear
functions (or basis functions). In this form we have a
system of N linear equations with N unknowns. This
can be assembled into a matrix equation which can be
solved, giving an approximate solution for each
element or degree of freedom.
BEM is also concerned with dividing the problem into
small elements, but it is slightly different in concept to
FEM. BEM deals with the PDE by reducing it to a
surface integral equation. The integral equation
determines the value of the independent variable on
the boundary surface due to contributions of sources
within the domain and the scattering effect of the
boundary itself. This provides enough information to
be able to solve the problem for any given point in the
domain. Since BEM works in this manner, it requires
only the boundary to be meshed rather than the full
domain, thus reducing the degrees of freedom in the
problem by an order of magnitude. BEM is not
suitable for all types of problems however since it is
limited to domains in which a Green’s function can be
calculated, which restrict it to linear and homogenous
media.
When using numerical methods for acoustics
involving meshing a domain, it is important to ensure
that the mesh is fine enough to accurately represent
the wave patterns. Opinion varies but it is typically
suggested that the mesh has a resolution of a
minimum 8 elements per maximum wavelength. For
a 3D mesh at full bandwidth, this can quickly lead to
models with degrees of freedom in the order of
millions. Due to computational expense, therefore, the
application of numerical methods in acoustics has only
recently been a practical option, and is still only really
feasible in small sized or low frequency problems.
3.3 Application of numerical methods to
headphone models
An ability to model wave behaviour in enclosed spaces
with adaptable geometry opens up the possibility of
full bandwidth 3D models of the sound fields under
headphones. Figure 20 demonstrates a crude
approximation of a coupling cavity mesh for a
circumaural headphone. However as discussed in
previous sections, headphones are more complex than
the simple acoustic enclosure, and hence any model
Figure 20: Example mesh of approximated
circumaural headphone coupling cavity
M.R AVIS AND L. J.KELLY
AUDIO AT HOME – AES 21ST
UK CONFERENCE 2006
10
must be able to accommodate additional mechanical
coupling.
As an example, consider the cushion. An elementary
numerical model may model the cushion as a locally
reacting surface and represent it as a known boundary
impedance, with data obtained from measurement.
This may be a reasonable approximation in some or
even most circumstances. However, the admittance
may not necessarily be locally reacting, there may be
some significant coupling of the cavity with the
cushion, or the cushion may be locally reacting but
may present an impedance that varies with position.
A more complete model of the headphone would
include a model of the cushion as a porous medium.
Indeed this may be the only option if the headphone
cushion extends to form a layer in front of the drive
unit (as is the case with many supra-aural models).
Several well-known approaches exist for modelling
acoustic waves in porous media, which develop the
problem with varying levels of rigour [7,8,9]. In
general though, these models effectively represent the
cushion as a lossy medium, characterised by its
(complex) density and phase velocity. It should be
noted that since BEM is limited to homogenous media
a coupled cavity-cushion model using this method
would not be possible.
Additionally, we might consider the headphone driver.
So far the driver has been considered as executing
pistonic motion. As has already been suggested, this
is an approximation, especially at high frequencies. A
thorough solution would include a structural model of
the driver which could be coupled to the cavity, but
this would be no trivial task.
A more convenient solution might ignore the coupling
effects and present the driver as a complex vibrating
surface which nonetheless retains constant velocity
behaviour. Vibration data could be provided from
measurement, necessitating laser interferometry, but
this poses some difficulties since the diaphragms in
question are small and frequently transparent.
Additionally, the light mass of the diaphragm suggests
that a constant velocity assumption could be
considerably in error.
4.0 CONCLUSIONS
The design of a circum-aural headphone incorporating
a dynamic driver has been described, using lumped
parameter techniques. It has been shown that coupler
volume affects the spectral response of the device, and
that the choice of an open or closed back approach has
significant influence on the affect of ‘leaks’, where the
cushion makes an imperfect seal to the side of the
head. The compliance of these cushions has been
shown to be of primary importance in the attenuation
of external sound fields through the cup, along with
the mass of the cup itself and the applied headband
tension.
With the pressure field under a headphone cup
becoming distributed at higher frequencies, a modal
decomposition has been shown to give a more realistic
representation than lumped parameters. For simple
geometry a modal approach is possible analytically but
for a headphone it is only feasible using numerical
methods. Headphone issues such as structural
coupling of the cushion and driver can be represented
in a numerical model opening the possibility for
headphone models which are accurate for the full
audio bandwidth.
5.0 REFERENCES
[1] Beranek, L.L. ‘Acoustics’, American Inst. of
Physics (New York 1986)
[2] Olson, H.F., ‘Elements of Acoustical
Engineering’, Van Nostrand (New York 1947)
[3] Borwick, J., ‘Loudspeaker and Headphone
Handbook’, 2nd Ed., Focal Press (Oxford 1994)
[4] Stimson, M.R., ‘The spatial distribution of
sound pressure within scaled replicas of the human ear
canal’, J.Acoust.Soc.Am. 78(5), 1985, pp1596-1602
[5] Kinsler, L.E., Frey, A.R. et al, ‘Fundamentals
of Acoustics’, 4th Ed., Wiley (New York 2000)
[6] Morse, P.M. & Ingard, K.U. ‘Theoretical
Acoustics’, (Princeton University Press, 1968)
[7] Delany, M.E., Bazley, E.N. Acoustical
properties of fibrous materials Applied Acoustics, 3
(1970) 105-16
[8] Biot, M.A. The theory of propagation of
elastic waves in a fluid-saturated porous solid. I. Low
frequency range. II. High frequency range.
J.Acoust.Soc.Am 28 (1956) 168-91
[9] Allard, J.F. ‘Propagation of Sound in Porous
Media’, Elsevier (London, 1993)
